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Sobolev Met Poincaré MaxPlanckInstitut fur Mathematik in den Naturwissenschaften Leipzig Sob olev met Poincare by Piotr Hajlasz and Pekka Koskela PreprintNr Sob olev met Poincare Piotr Ha jlasz and Pekka Koskela Contents Intro duction What are Poincare and Sob olev inequalities Poincare inequalities p ointwise estimates and Sob olev classes Examples and necessary conditions Sob olev typ e inequalities by means of Riesz p otentials Trudinger inequality A version of the Sob olev emb edding theorem on spheres RellichKondrachov Sob olev classes in John domains John domains Sob olev typ e inequalities Poincare inequality examples Riemannian manifolds Upp er gradients Top ological manifolds Gluing and related constructions Further examples CarnotCaratheo dory spaces CarnotCaratheo dory metric Upp er gradients and Sob olev spaces Carnot groups Hormander condition Further generalizations Graphs Applications to PDE and nonlinear p otential theory Admissible weights Sob olev emb edding for p App endix Measures Uniform integrability p p spaces L and L w Covering lemma Maximal function Leb esgue dierentiation theorem Abstract There are several generalizations of the classical theory of Sob olev spaces as they are necessary for the applications to CarnotCaratheo dory spaces sub el liptic equations quasiconformal mappings on Carnot groups and more general Lo ewner spaces analysis on top ological manifolds potential theory on innite graphs analysis on fractals and the theory of Dirichlet forms The aim of this pap er is to present a unied approach to the theory of Sob olev spaces that covers applications to many of those areas The variety of dierent areas of applications forces a very general setting We are given a metric space X equipp ed with a doubling measure A generalization of a Sob olev function and its gradient is a pair u L X lo c p g L X such that for every ball B X the Poincaretyp e inequality Z Z p p ju u j d Cr g d B B B holds where r is the radius of B and C are xed constants Working in the ab ove setting weshow that basically all relevant results from the classical theory have their counterparts in our general setting These include Sob olev Poincare typ e emb eddings RellichKondrachov compact emb edding theorem and even a version of the Sob olev emb edding theorem on spheres The second part of the pap er is devoted to examples and applications in the ab ovementioned areas This research was b egun while PH was visiting the Universities of Helsinki and Jyvaskyla in continued during his stay in the ICTP in Trieste in and nished in MaxPlanck Institute MIS in Leipzig in He wishes to thank all the institutes for their hospitality PH was partially supp orted by KBN grantno POA PK by the Academy of Finland grant SA and by the NSF Mathematical Subject Classication Primary E Intro duction The theory of Sob olev spaces is a central analytic to ol in the study of various asp ects of partial dierential equations and calculus of variations However the scop e of its appli cationsismuch wider including questions in dierential geometry algebraic top ology complex analysis and in probability theory p Let us recall a denition of the Sob olev spaces Let u L where isanopen n p subset of IR and p We say that u belongs to the Sob olev space W if p the distributional derivatives of the rst order b elong to L This denition easily extends to the setting of Riemannian manifolds as the gradientiswell dened there The fundamental results in the theory of Sob olev spaces are the socalled Sob olev emb edding theorem and the RellichKondrachov compact emb edding theorem The p p rst theorem states that for p n W L where p npn p provided the b oundary of is suciently nice The second theorem states that for p q every qp the emb edding W L is compact Since its intro duction the theory and applications of Sob olev spaces have b een under intensive study Recently there have b een attempts to generalize Sob olev spaces to the setting of metric spaces equipp ed with a measure Let us indicate some of the problems that suggest such a generalization Study of the CarnotCaratheo dory metric generated by a family of vector elds Theory of quasiconformal mappings on Carnot groups and more general Lo ewner spaces Analysis on top ological manifolds Potential theory on innite graphs Analysis on fractals Let us briey discuss the ab ove examples The CarnotCaratheo dory metric app ears in the study of hyp o elliptic op erators see Hormander Feerman and Phong Jerison Nagel Stein and Wainger Rotschild and Stein SanchezCalle The Sob olev inequality on balls in CarnotCaratheo dory metric plays a crucial role in the socalled Moser iteration technique used to obtain Harnack inequalities and Holder continuity for solutions of various quasilinear degenerate equations The pro of of the Harnack inequalityby means of the Moser technique can b e reduced to verifying a suitable Sob olev inequality Conversely a parab olic Harnack inequality implies a version of the Sob olev inequality as shown by SaloCoste It seems that the rst to use the Moser technique in the setting of the CarnotCaratheo dory metric were Franchi and Lanconelli The later work on related questions include the pap ers by Biroli and Mosco Buckley Koskela and Lu Cap ogna Danielli and Garofalo ChernikovandVodopyanov Danielli Garofalo Nhieu Franchi Franchi Gallot and Wheeden Franchi Gutierrez and Wheeden Franchi and Lanconelli Franchi Lu and Wheeden Franchi and Serapioni Garofalo and Lanconelli Marchi The theory of CarnotCaratheo dory metrics and related Sob olev inequalities can b e extended to the setting of Dirichlet forms see Biroli and Mosco Garattini Sturm For connections to the theory of harmonic maps see the pap ers Jost Jost and Xu Ha jlasz and Strzelecki The theory of quasiconformal mappings on Carnot Groups has b een studied by Margulis and Mostow Pansu Koranyi and Reimann Heinonen and Koskela Vo dopyanov and Greshnov Results on Sob olev spaces play an imp ortant role in this theory Very recently Heinonen and Koskela extended the theory to the setting of metric spaces that supp ort atyp e of a Sob olev inequality Semmes has shown that a large class of top ological manifolds admit Sob olev typ e inequalities see Section Sob olev typ e inequalities on a Riemannian manifold are of fundamental imp ortance for heat kernel estimates see the survey article of Coulhon for a nice exp osition Discretization of manifolds has lead one to dene the gradient on an innite graph using nite dierences and then to investigate the related Sob olev inequalities see Kanai Auscher and Coulhon Coulhon Coulhon and Grigoryan Delmotte Holopainen and Soardi These results have applications to the classication of Riemannian manifolds Also the study of the geometry of nitely generated groups leads to Sob olev inequalities on asso ciated Cayley graphs see Varop oulos SaloCoste and Coulhon and Section for references At last but not least the Brownian motion on fractals leads to an asso ciated Laplace op erator and Sob olev typ e functions on fractals see Barlow and Bass Jonsson Kozlov Kigami Kigami and Lapidus Lapidus Metz and Sturm Mosco How do es one then generalize the notion of Sob olev space to the setting of a metric space There are several p ossible approaches that we briey describ e below In general the concept of a partial derivative is meaningless on a metric space However it is natural to call a measurable function g an upp er gradient of a function u if Z jux uy j gds holds for each pair x y and all rectiable curves joining x y Thus in the Euclidean setting we consider the length of the gradient of a smo oth function instead of the actual gradient The ab ove denition is due to Heinonen and Koskela Assume that the metric space is equipp ed with a measure Then we can ask if for every pair u g where u is continuous and g is an upp er gradientofutheweak version Z Z q p q p ju u j d g d Cr B B B of the Sob olevPoincare inequality holds with qp whenever B is a ball of radius r Here C and are xed constants barred integrals over a set A mean integral averages and u is the average value of u over B B If q we call a pPoincare inequality It turns out that a pPoincare inequality implies a Sob olevPoincare inequalitysee Section This approachishowever limited to metric spaces that are suciently regular There have to b e suciently many rectiable curves which excludes fractals and graphs For more information see Section Section Section Bourdon and Pa jot Cheeger Franchi Ha jlasz and Koskela Hanson and Heinonen Heinonen and Koskela Kallunki and Shanmugalingam Laakso Semmes Shanmugalingam Tyson Recently Ha jlasz intro duced a notion of a Sob olev space in the setting of an arbitrary metric space equipp ed with a Borel measure that we next describ e n p One can prove that u W p where IR is a b ounded set p with
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